Lucas' theorem refined

Hwa Long Gau, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

Abstract

We prove a refined version of the classical Lucas' theorem: if p is a polynomial with zeros a1,...,an+1 all having modulus one and φ is the Blaschke product whose zeros are those of the derivative p′, then the compression of the shift S(φ) has its numerical range circumscribed about by the (n + 1)-gon a1...an+1 with tangent points the midpoints of the n + 1 sides of the polygon. This is proved via a special matrix representation of S(φ) and is a generalization of the known case for n = 2.

Original languageEnglish
Pages (from-to)359-373
Number of pages15
JournalLinear and Multilinear Algebra
Volume45
Issue number4
DOIs
StatePublished - 1999

Keywords

  • Compression of the shift
  • Dilation
  • Numerical range

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